LINEAR SYSTEMS of INEQUALITIES – Graphing Method

When graphing a set of linear inequalities, the solution set will be a “shared” space where the two solutions intersect.

You can see where the tan and gray have a common area where they are “on top” of each other.

Coordinates in that shared space will satisfy BOTH inequalities. SHARED

AREA

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

When graphing a set of linear inequalities, the solution set will be a “shared” space where the two solutions intersect.

You can see where the tan and gray have a common area where they are “on top” of each other.

Coordinates in that shared space will satisfy BOTH inequalities.

Once again you can use either slope – intercept or an ( x , y ) table to graph.

SHARED AREA

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

When graphing a set of linear inequalities, the solution set will be a “shared” space where the two solutions intersect.

You can see where the tan and gray have a common area where they are “on top” of each other.

Coordinates in that shared space will satisfy BOTH inequalities.

Once again you can use either slope – intercept or an ( x , y ) table to graph.

When graphing your lines, use

Dashed line for : < or >

Solid line for : ≤ or ≥

SHARED AREA

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 - 2

2

63

6302

632

y

y

y

yx

I replace the inequality with an equal sign when getting my ( x , y ) points. That way I don’t get confused when solving…

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 - 2

3 -4

4

123

636

6332

632

y

y

y

y

yx

I replace the inequality with an equal sign when getting my ( x , y ) points. That way I don’t get confused when solving…

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 - 2

3 -4

Since the equation has a ( < ) symbol, use a dashed line to graph…

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 - 2

3 -4 To find the side to shade, use a test point to find TRUE or FALSE…

ALWAYS shade the TRUE side…

I like to use ( 0 , 0 )

60

600

60302

TEST POINT

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 - 2

3 -4 Since the test point is FALSE, we will shade the other side that does not contain the test point…

FALSE

60

600

60302

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 - 2

3 -4

60

600

60302

Since the test point is FALSE, we will shade the other side that does not contain the test point…

FALSE

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 5

Let’s get our other set of points to graph…

5

503

53

y

y

yx

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 5

1 8

Let’s get our other set of points to graph…

8

53

513

53

y

y

y

yx

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 5

1 8

Let’s get our other set of points to graph…

Since the equation has a ( ≥ ) symbol, use a solid line to graph…

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 5

1 8

Again, use ( 0 , 0 ) as a test point…

50

500

5003

53

yx

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 5

1 8

Again, use ( 0 , 0 ) as a test point…

FALSE !!! So shade the side that does not contain the test point…

50

500

5003

53

yx

FA

LS

E

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

x Y

0 5

1 8

Again, use ( 0 , 0 ) as a test point…

FALSE !!! So shade the side that does not contain the test point…

50

500

5003

53

yx

FA

LS

E

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

You can see where the two colors overlap…

That is the solution area.

ANY coordinate point in that “shared area” will satisfy both inequalities…

SHARED

AREA

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 1 :

53

632

yx

yx

Let’s check the point ( - 6, - 2 ) …

618

6612

62362

632

yx

( - 6, - 2 )

516

5218

5263

The check coordinate satisfy’s BOTH …

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 2 :

yx

yx

43

2

102

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 2 :

Let’s use slope – intercept this time…I still replace the inequality with an equal sign when solving for “y”…

52

1

102

102

xy

xy

yx

yx

yx

43

2

102

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 2 :

Let’s use slope – intercept this time…I still replace the inequality with an equal sign when solving for “y”…

52

1

102

102

xy

xy

yx

yx

yx

43

2

102

52

1

b

m

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 2 :

Let’s use slope – intercept this time…I still replace the inequality with an equal sign when solving for “y”…

yx

yx

43

2

102

Equation has a ≥ sign so use a solid line…

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 2 :

Let’s use slope – intercept this time…I still replace the inequality with an equal sign when solving for “y”…

100

10020

102

yx

yx

yx

43

2

102

Test point ( 0 , 0 ) is FALSE…

FALSE

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 2 :

Let’s use slope – intercept this time…I still replace the inequality with an equal sign when solving for “y”…

100

10020

102

yx

yx

yx

43

2

102

Shade the opposite side…

FALSE

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 2 :

yx

yx

43

2

102

43

2

b

m

This is already in y = mx + b form…

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 2 :

yx

yx

43

2

102

43

2

b

m

Equation has a ≥ sign so use a solid line…

This is already in y = mx + b form…

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 2 :

yx

yx

43

2

102

Test point ( 0 , 0 ) is TRUE…so shade the side with the test point…

04

040

0403

2

TRUE

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 2 :

yx

yx

43

2

102

SHARED

AREAYou can see where the two

colors overlap…

That is the solution area.

Again, ANY coordinate point in that “shared area” will satisfy both inequalities…

LINEAR SYSTEMS of INEQUALITIES – Graphing Method

EXAMPLE # 2 :

yx

yx

43

2

102

Let’s check the point ( 6 , - 4 )

( 6 , - 4 )

1014

1086

10426

102

yx

48

444

4463

2

The check coordinate satisfy’s BOTH …